Stochastic evolution equations with a spatially homogeneous Wiener process
A Cauchy problem for a stochastic parabolic equation \[ \frac {\partial X}{\partial t} = AX + f(t,x,X) + b(t,x,X) \dot W, \quad X(0)=\zeta , \tag{1} \] in \(\mathbb{R}^{d}\) is considered. It is assumed that \(A\) is a uniformly elliptic \(2m\)th order differential operator with bounded smooth coefficients, and the functions \(f(t,x,\cdot)\), \(b(t,x,\cdot)\) satisfy the Lipschitz and linear growth conditions uniformly in \((t,x)\). \(W\) denotes a Wiener process in the space \(\mathcal S'\) of tempered distributions on \(\mathbb{R}^{d}\) which is supposed to be spatially homogeneous, that is, the law of \(W(t)\) is invariant with respect to all translations of \(\mathbb{R}^{d}\). It is known that the covariance form of such a process is given by a distribution \(\Gamma \in \mathcal S'\) which is a Fourier transform of a Borel positive and symmetric measure \(\mu \) on \(\mathbb{R}^{d}\). The equation (1) is dealt with by means of the abstract semigroup approach to stochastic partial differential equations. Sufficient conditions on the spectral measure \(\mu \) are found under which (1) has a unique mild solution in a suitable weighted \(L^2\)-space on \(\mathbb{R} ^{d}\). Moreover, (1) is shown to define a Feller Markov process in this space. Further, regularity of solutions in the space variable is investigated. Finally, it is shown that solutions to (1) with a diffusion coefficient \(\varepsilon b\), \(\varepsilon >0\), obey a large deviations principle. Notwithstanding that equations analogous to (1) have been already treated several times starting with the paper by \textit{D. A. Dawson} and \textit{H. Salehi} [J. Multivariate Anal. 10, 141-180 (1980; Zbl 0439.60051)], the paper under review provides both finer and more general results and new methods that are likely to be very useful in related problems.
scientific article; zbMATH DE number 1409157
scientific article